Sunday, February 26, 2017

The case for string theory: 60 symbols

Tony Padilla is a cosmologist who has recorded numerous successful videos about science and mathematics. Many of them showed some interesting mathematics, some of the videos on the same channel (not connected with Tony) were embarrassing (e.g. the video "quantum mechanics, an embarrassment").

Three weeks ago, he released this 16-minute video at the "Sixty symbols" channel (OK, it's really "SIXTΨ SγMBΦLS", I had to waste a minute by writing these symbols; fortunately, fewer than 60 were needed). It already has over 150,000 views and 98% of the votes are positive.

Padilla reveals himself – like Leonard Hofstadter did – to be in the string camp. His mature justification is that he sees the natural progression towards the string concepts that is analogous to other successful developments in the history of physics, including the closely analogous route that led to the discovery of the Higgs boson.

Instead of focusing on the usual stuff – what strings look like, how they dance, how they require extra dimensions in their apartment buildings – he focuses on the consistency of the particle interactions and the analogy between the Higgs boson and strings. You need to describe a force. Within relativistic quantum mechanics, it's unavoidably linked to the exchange of a virtual particle. You need to give this particle some properties but the combination of relativity and quantum mechanics is highly constraining and tells you that you have to do things right, basically in a unique way.

When you try to explain the beta-decay, you find out that a four-fermion interaction is needed and to be renormalizable, a spin-one boson – the W-boson – has to be mediating it. However, when you study the possible interactions of these bosons with themselves, you find out that their masses can't be just explicitly given. They must result from a broken Yang-Mills theory i.e. from the Higgs mechanism which also implies the existence of a new particle, the Higgs boson.

Mathematically, the existence of the new Higgs boson may also be explained as the existence of poles of the scattering amplitudes. These singular points have to be there by some "complex calculus argument".

Analogously, in string theory, you first find out that gravity has to be mediated by the exchange of gravitons. But like the four-fermion interaction, the strength of this interaction would get out of control at high energies so there must be some cancelling contributions to the scattering amplitudes. These new poles have to be arranged in a way that obeys some scaling laws and they match the spectrum of a string. So the exchange of a graviton is just the exchange of a massless excitation of a string, but other states of the strings are also exchanged and that's what guarantees the UV finiteness. These newly predicted particles play a bonus role: they may be viewed as the perturbative string progenitors of the black hole microstates – whose density increases exponentially (well, exponential of a power) with the mass.

So all these extra degrees of freedom that the string-like character of the particles produce are needed for the UV finiteness of the amplitudes as well as for the incorporation of the building blocks for black hole thermodynamics. Perturbative string theory isn't the "only" way to skin the cat – but string theory is. Perturbative string theory is the only known way to skin the cat that simultaneously allows perturbative expansions in a coupling constant, at least for quantum gravity in \(D\geq 4\).

Padilla says that people didn't push strings – "let's try strings" – from the beginning. Instead, they were led to various formulae, like the Euler Beta function, and then they simply interpreted the formulae. Look, the formula we had describes strings. Well, right, it's what happened but it happened just a few years after the stringy formulae first emerged. Afterwards, for over 40 years, people were rather deliberately "studying the physics of strings". Padilla is right that string theory isn't just some "random childish idea" to replace the point-like particles by snowflakes or little green men.

But I don't really think that it would be a "sin" – or something that seriously reduces the value of string theory – if it started by the slogan "let's try strings". I do think that people should have tried strings and other extended objects many years before they actually did so. Well, Dirac and others actually tried to describe particles as extended objects but they didn't do it right. The theories on the two-dimensional world sheet are amazingly consistent and people were forced to realize this fact only after they learned about the Veneziano amplitude and other mathematical results from "other sources", sources that don't assume extended objects.

With hindsight, I do believe that it was extremely narrow-minded and prejudiced that people kept on assuming that the fundamental objects had to be point-like for such a long time (and, similarly, why they were almost universally assuming that there weren't extra compactified dimensions etc.). They were just restricting the space of possible candidate ideas by unnatural constraints. These opinions clearly depend on one's subjective taste and there's no God-given "unique right way how a theory should be discovered". But I would insist that string theory is natural not only from the perspective that Padilla has understood and is successfully selling.

In the second part, he says that the large number of vacuum-like solution may be a weakness or a virtue, depending on one's viewpoint, and he argues it's not true that string theory isn't testable. Some words about the cosmological constant, supersymmetry, and possible proofs of uniqueness may be heard, too.